Optimal path planning of Unmanned Aerial Vehicles (UAVs) for targets touring: Geometric and arc parameterization approaches

The path planning problem for unmanned aerial vehicles (UAVs) is important for scheduling the UAV missions. This paper presents an optimal path planning model for UAV to control its direction during target touring, where UAV and target are at the same altitude. Geometric interpretation of the given model is provided when the vehicles consider connecting an initial position to the destination position with specific target touring. We develop a nonlinear constrained model based on an arc parameterization approach to determine the UAV’s optimal path touring a target. The model is then extended to touring finite numbers of targets and optimizing the routes. The model is found reliable through several simulations. Numerical experiments are conducted and we have shown that the UAV’s generated path satisfies vehicle dynamics constraints, tours the targets, and arrives at its destination.

Response. Thank you for pointing this out. Our experiments are two-fold. We first model the problem in AMPL and solve the problem using efficient solvers which has been reported in the manuscript. Once we get the solutions, we use MATLAB to visualize the output.
Therefore, in the revised version, we included the total time elapsed by the solver, total time taken by the system and total visualization time for each of our experiments. The

Response.
Our analysis related to obtaining the shortest path in 2D environment with the simulation demonstrated widely depends on the 2-dimensional coordinates system. To make some tests in the real world and in our current simulation to validate the usefulness of our work in real robot, we need to extend the current analysis in 3D coordinate system that requires theoretical and algorithmic development. We appreciate Editor's suggestion and keep this implementation as our future research work. 2 Comment 1.5 Please, extend the explanation about how you will deal with obstacle avoidance in a future work, give more details.
Response. We have incorporated the reviewer's suggestions and described how our proposed model could be extended to obstacle avoidance.
We have included the following clarifications on Page 15 with slight adjustments in the revised manuscript.
The material is presented in this paper can be extended for the obstacle avoidance problem.
UAVs fly through obstacles such as buildings, hills, restricted zones, etc., which intercept the normal flight paths of the UAV. Sensors monitoring the UAV's environment for fixed or moving obstacles are usually used for obstacle avoidance. The obstacle avoidance problem is closely associated with path planning because obstacles typically result in the re-planning of paths. For obstacle avoidance, our proposed model requires slight changes in the geometrical approach given in Fig (1) and then requires expressing the mathematical functions to construct the flying paths. For example, suppose the obstacle is circular, in that case, the target point T must be on the circumference of the minimum turning circle centred at C (see Fig 3   ). The point T is variable here, and it is required to find T in such a way so that the flying path is minimum.

Reviewer 1 Comments
Comment 2.6 Figure 1 and its description should be explained in a little more detail since the interpretation is confusing.
Response. We have incorporated the reviewer's suggestions and described Fig 1 in further detail.
We have included the following clarifications on Page 3 with slight adjustments in the revised manuscript.

Geometrical Interpretation Path Planning for Single Target Touring
This section illustrates path planning for UAV that start from an initial location, passes through a known target, and reach the finishing point. The proposed geometric approach describes an essential criterion for shortest route planning subject to the conditions that UAV tour the target. Our proposed geometrical interpretation of a feasible optimal path of type CSCSC for a single target is the concatenation of two sub-paths of type CS and CSC. It is noted that initial configuration (a configuration consists of a position and a heading angle) P I (x 0 , y 0 , ψ 0 ) and the final configuration is P f (x f , y f , ψ f ). The fixed target location is T (x T , y T , ψ T ), which is known in advance where ψ T is the angle at T formed by straight lines X 1 T or X 2 T presented in (1). According to the Fig (1), UAV moves from P I with a given heading angle ψ 0 and then flies towards target T, and after touring T, it turns along its destination P f .
We divide the minimum path design for target touring problems into following two steps, as shown in Fig (1).
Comment 2.7 Equation 3 should be revised.

Response. We have incorporated the reviewer's suggestions. Now Equation (3) of old manuscript has been revised and referred to as Problem (P).
We have included the following clarifications on Page 5 with slight adjustments in the revised manuscript.
Response. We thank the reviewer for pointing this out. We have rewritten Equations (5), (6), (7), (8), and (9) and used standard math text. We also used proper notations to write "sin, arg, cos, for all" which can be seen on Page 6 of the revised manuscript. 2 Comment 2.9 The authors can enrich the explanation of section 4 using algorithms or flow charts.
Response. We have incorporated the reviewer's suggestions. In addition, we have included an algorithm in Section 4 to explain the numerical steps of our implementation.
We have included the following algorithm on Page 7 with slight adjustments in the revised manuscript.
Now we adopt mathematical approaches to construct the model for finding the optimal path of UAV with single target touring describe as follows.
Step 2 (Determine the length from initial point to target point) Step 3 Find optimal path that solves Problem (P):= min(CS + CSC).
To determine optimal path for single target touring, we combine the feasible paths obtained in Steps 2 and 3 and thus optimal path be one of the feasible paths listed in (11). For instance, if we obtain optimal of type RSLSR, then we attained lengths with l 1 = l 5 = l 7 = 0 and l 2 , l 3 , l 4 , l 6 , l 8 > 0.
works. In addition, it is essential that the authors place the computation time and the amount of memory spend to obtain each result.
Response. We have incorporated the reviewer's suggestions and conducted further experiments to compare our proposed model (26)

Reviewer 2
Comment 3.1 The paper is interesting, and presents a congruent mathematical analysis. I consider that the use of the drone at the same height should be justified in a solid way, as this is a point of improvement to the proposed analysis. I also consider that more experiments with radii of circumferences smaller than 2 should be shown.

Response.
We appreciate the reviewer's valuable comments and would like to thank you for finding our analysis interesting. In our analysis, we use 2D path and propose a new model to find the solution to the arc lengths so that we can identify the shortest path from the possible feasible solutions described by Dubin's [1] (reference of the revised manuscript).
We have now conducted more experiments which can be seen in Comment 1.1. We also conducted experiments taking smaller radii, and the following experiment has now been included with slight adjustment on Page 11 of the revised manuscript.
We also solved Example 2 considering with smaller radii r = 1, and obtained the shortest path depicted in Figure 4. Optimal solutions for the arc lengths are included in Table 2